Fine regularity of stochastic processes is usually measured in a local way by local
Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter
Gaussian random fields, Adler proved that these two concepts are connected under the
assumption of increment stationarity property. The aim of this paper is to consider the
case of Gaussian fields without any stationarity condition. More precisely, we prove that
almost surely the Hausdorff dimensions of the range and the graph in any ball
B(t0,ρ)
are bounded from above using the local Hölder exponent at t0. We define
the deterministic local sub-exponent of Gaussian processes, which allows to obtain an
almost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of the
sample path on an open interval are controlled almost surely by the minimum of the local
exponents. Then, we apply these generic results to the cases of the set-indexed fractional
Brownian motion on RN, the multifractional
Brownian motion whose regularity function H is irregular and the generalized Weierstrass
function, whose Hausdorff dimensions were unknown so far.